DEVELOPMENT OF 2- AND 3-D SIMULATOR FOR THREE-PHASE FLOW WITH GENERAL INITIAL AND BOUNDARY CONDITIONS ON THE FRACTIONAL FLOW APPROACH

ABSTRACT

 

 

This thesis presents the development of 2-D and 3-D multiphase flow simulators as named 2DMPS(2–DMultifluid Phase Simulator) and 3DMPS (3–DMultifluid Phase Simulator) for three-phase flow (Water, NAPL, and Gas) in the subsurface, respectively. The 2DMPS and 3DMPS are developed using the hybrid Lagrangian and Eulerian approach (e.g. LEZOOMPC, The Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture scheme) and the Eulerian approach (e.g.

Galerkin upstream finite element method).

 

The governing equations of the fractional flow based approach consist of two saturation equations having an advection-diffusion form with advection-dominated term, and a global pressure equation having an elliptic form. The former set of equations is well suited for numerical solution with the method of characteristics while the latter can be solved with conventional finite element method (e.g. Galerkin finite element method) Accordingly, a mixed Lagrangian-Eulerian approach (LEZOOMPC) is used for solving simultaneously two coupled nonlinear saturation equations, in which the advection term can be effectively and accurately dealt with the LEZOOMPC algorithm.  Specifically, the accuracy in determining the Lagrangian saturation in most Lagrangian-Eulerian methods including LEZOOMPC depends on both the particle tracking algorithm and interpolation scheme. Most particle tracking methods are limited for the steady state flow case. However, in this research a multidimensional particle tracking algorithm has been

 

developed to account for both temporal and spatial variations of velocity fields during the time step and incorporated into the LEZOOMPC algorithm.

 

In addition to the aforementioned ability of effectively and accurately solving the fractional flow approach equations with the LEZOOMPC scheme, fractional flow approach has its own inherent advantages. Since primary variables such as water and total liquid saturations are adopted in the fractional flow approach, 2DMPS and 3DMPS have the capability of automatically handling phase change configuration during the simulation time without any assumption or variable switching technique. The primary variables can be always defined no matter how a set of phase is changed in both time and space. Even though the fractional flow based approach has the several advantages over the pressurebased approach, it also has a difficulty in dealing with boundary conditions that are very often encountered in groundwater literature. However, in this research, general initial and boundary conditions are incorporated into 2DMPS and 3DMPS. The general boundary condition consists of ten types. Eight of these are combinations of the flux type or Dirichlet-pressure type of each individual phase and the last two, defined as variable boundary conditions, are the combination of gradient of capillary pressure or flux type of each individual phase depending on the velocity direction of each phase on the boundary. The general initial condition consists of eight types, which are the combinations of initial pressure and saturation of three individual phases. Any type of aforementioned initial and boundary conditions have been transformed and incorporated in terms of the primary variables, a global pressure and two saturations.

 

In this study, 2DMPS and 3DMPS are developed and verified with an analytical solution and other numerical software (e.g. 2DFATMIC). Also several examples are presented to represent the treatment of general boundary and initial conditions and automatic adaptation of phase appearance and disappearance without any variable switching technique, and to show the applicability to real or field problems. Also an algorithm of multidimensional particle tracking technique accounting for transient state flow is presented and verified with analytical solution for both accuracy and efficiency.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE OF CONTENTS

 

 

 

LIST OF FIGURES……………………………………………………………………..viii

 

LIST OF TABLES…………………………………………………………….…………xiv

 

ACKNOWLEDGEMENTS……………………..………………………………………..xv

 

CHAPTER 1. INTRODUCTION…………………………………………………………1

1.1.MOTIVATION AND OBJECTIVES…………………………………………………2

• BACKGROUND………………………………………………………………………5

1.3. RESEEARCH OUTLINE…………………………………………………………….9 1.4. FORMAT……………………………………………………………………………11

REFERENCES…………………………………………………………………………..13

 

CHAPTER 2. MULTIPHASE FLOW MODELING WITH GENERAL BOUNDARY

CONDITIONS AND PHASE CONFIGURATION CHANGE USING FRACTIONAL

FLOW APPROACH……………………………………………………………………..19

ABSTRACT………………………………………………………………………20

• INTRODUCTION…………………………………………………………..21
• GOVERNING EQUATIONS……………………………………………….24
• NUMERICAL FORMULATION OF MULTIPHASE FLOW IN

FRACTIONAL FLOW APPROACH………………………………………42

• VERIFICATION…………………………………………………………….49
• RESULTS AND DISCUSSIONS…………………………………………..54
• CONCLUSIONS……………………………………………………………68

ACKNOWLEDGEMENTS………………………………………………………69

REFERENCES……………………………………………………………………70

 

CHAPTER 3. PARTICLE TRACKING ALGORITHM FOR THE LAGRANGIANEULERIAN FINITE ELEMENT METHODE UNDER THE TRANSIENT

CONDITIONS IN MULTI-DIMENSIONS……………………………………………116

ABSTRACT…………………………………………………………………….117

• INTRODUCTION…………………………………………………………118
• PARTICLE TRACKING PROCESS IN TRANSIENT STATE

SIMULATION……………………………………………………………..121

• PARTICLE TRACKING ALGORITHM………………………………….126
• NUMERICAL EXPERIMENTS…………………………………………..170
• CONCLUSIONS…………………………………………………………..180

APPENDIX……………………………………………………………………..182

REFERENCES…………………………………………………………………184 CHAPTER 4. THREE-DIMENSIONAL THREE-PHASE FLOW SIMULATIONS USING THE LAGRANGIAN-EULERIAN APPROACH WITH ADAPTIVELY

ZOOMING AND PEAK/VALLEY CAPTURING SCHEME (LEZOOMPC)………..195

ABSTRACT…………………………………………………………………….196

• INTRODUCTION…………………………………………………………197
• MATHEMATICAL MODEL………………………………………………199
• APPLICATION OF LEZOOMPC APPROACH………………………….212
• VERIFICATION…………………………………………………………..218
• RESULTS AND DISCUSSIONS………………………………………….222
• CONCLUSIONS…………………………………………………………..233

REFERENCES…………………………………………………………………234

 

CHAPTER 5. SUMMARY AND FUTURE WORK…………………………………..276

• DIFFICULTIES OF APPLICATION OF THE DEVELOPED MODELS IN

THE LABORATORY OR FIELD…………………………………………277

• SUMMARY………………………………………………………………..279
• FUTURE WORK…………………………………………………………..281

CHAPTER 1.

 

INTRODUCTION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.1 MOTIVATION AND OBJECTIVES

 

The quantitative evaluation of contamination of ground water by immiscible fluids is an area that only recently has received much attention. The increased attention results from previous waste disposal practices and accidental spills, which have resulted in the placement of large quantities of immiscible fluids in or above ground water. Furthermore, this attention is much greatly enhanced since general contaminant transport models for miscible components can not describe the migration of the immiscible contaminant because the flow of an immiscible contaminant is controlled by its own flow potential, which depends on pressure, gravity, and surface forces, and is not necessarily similar to the ground water flow potential. Thus since the 1980’s an increasing number of problem in environmental engineering, mainly dealing with groundwater protection, remediation or groundwater management, have proved to require numerical simulations.

 

Many researchers in the hydrologic literature have been involved in developing numerical simulators for modeling multiphase flow using the pressure-based approach. On the other hand, in the petroleum engineering literature, multiphase flow simulators had been popularly developed using the fractional-flow based approach. In the pressure based approach, individual balance equations for each of the fluids are used to solve multiphase flow problems, while fractional flow based approach involves manipulation and combination of those balance equations into modified forms, with concomitant introduction of ancillary functions such as the fractional flow function. The pressure approach has been widely used in the hydrologic literature. In this approach, the governing equations are written in terms of the pressures through a straightforward substitution of Darcy’s equation into the mass balance equations for each phase. This approach has been adopted by a number of authors, [3, 5, 7, 15, 23, 25, 26, 34, 41, 46] .  In the pressure-based approach, governing equations for individual phases are strongly coupled and each is a nonlinear mixed hyperbolic-parabolic equation. In the fractional-flow based approach, the introduction of global pressure produced an elliptic and two hyperbolic-dominant equations.  As a result, the fractional-flow based approach offers some advantages over the pressure-based approach. First, the system of equation is almost decoupled and the elliptic equation is almost linear. Second, the hyperbolic-dominant equation can be effectively solved with innovative convective algorithms, e.g., Lagrangian-Eulerian approaches with adaptively local grid refinement and peal/valley capturing schemes. Finally, in the pressure-based approach, change of phase configuration might cause some errors in the solution because of introduction of cumbersome error tolerance for variable switching technique or specification of negligible saturation where phase is absent in the domain [8, 48] while since in the fractional flow based approach all primary variables can be always defined regardless of whether some phases are present or not, phase configuration change can be automatically considered with keeping governing equation as same form.

 

Even though the fractional flow based approach has the inherent advantages over the pressure based approach as mentioned before, it still has the disadvantage over the pressurebased approach. One disadvantage is that it is difficult to manage general boundary conditions commonly appeared in hydrology compared to the pressure based approach. In petroleum reservoirs, where the total flux is often adopted as a boundary condition, fractional flow approach has no difficulty in using this boundary condition. However in hydrology boundary conditions are often posed in terms of individual fluid fluxes or pressures. While pressure-based approaches can easily handle these types of boundary conditions, fractional flow-based approaches cannot directly use them to solve the multiphase problem, because they cannot be explicitly transformed into boundary conditions for such approaches except for a few specific cases.  Fractional flow-based approaches require that the boundary conditions be posed as the total pressure or total flux for the total-pressure equation, the water saturation or the water flux for the water saturation equation, and the liquid saturation or the liquid flux for the liquid saturation equation.

 

The fractional-flow based approach has been employed by several authors [4, 21] . However, until now most fractional flow approaches have been limited to two phases and specific boundary conditions. According to Binning and Celia [4] , they used fractional flow approach to aim at gaining significant improvement in computational efficiency, but they pointed out that when generalized boundary conditions are incorporated into model, there was little or no computational advantage because fractional flow equation must be coupled with nonlinear iterative scheme to find the correct boundary conditions. Thus, they considered the only one boundary type, which is a flux type for water phase, and Dirichlet type for air phase.

 

 

The objective is to develop a numerical simulator for migration of multi-dimensional three phases flow (water, NAPL, and gas) using the fractional-flow based approach to realize some advantages over the pressure-based approach. Accordingly, in order to use nature of hyperbolic-dominant partial differential equation, we use advanced Lagragian-Eulerian methods to solve saturation equation such as LEZOOMPC [49] . Also, for the model’s flexibility and efficiency to handle general boundary condition, robust and efficient strategies are developed to iteratively (using an outer iteration loop) transform general types of boundary conditions for pressure-based approaches to those for fractional flow-based approaches.

 

 

1.2. BACKGROUND

 

There has been a significant progress on multiphase flow modeling during the last two decades in ground water literature and for the last three decades in petroleum reservoir engineering. Development of the governing equations describing multiphase flow and multicomponent transport in a porous medium dates back to the early 1960s, when the petroleum industry was interested in quantifying production and in evaluating the efficiency of secondary and tertiary oil recovery methods. The first solutions of the compositional system for petroleum reservoir systems were published in the 1960s. Welge et al. [45] extended the Welge [44] technique for solving two-phase Buckley and Leverett [6] flow to simulate one-dimensional condensing gas drive. The new technique accounted for two-phase (oil-gas) flow with interphase mass transfer of three components. Simplified numerical models of one- and two- dimensional flow and transport with interphase mass transfer were later developed by Price and Donohue [35] and Van-Quy et al. [43] . More recently, very comprehensive field scale models incorporating simultaneous water, gas, and oil flow and transport, heat transfer, and complex phase relationships have been used to evaluate the feasibility of enhanced oil recovery processes such as steam flooding, in situ combustion, and surfactant and polymer flooding (see, for example, Coats [12] , Rubin and Buchanan [40] , Gupta et al. [22] , and Quandalle and Sabathier [39] ). These comprehensive reservoir simulations are for the most part not directly applicable to the simulation of organic contamination of groundwater systems. The majority of the reservoir simulators assume no partitioning of organic to the water phase. Capillary pressures, dispersion, and diffusion are often neglected as they are not importance at the scale of discretization typical of petroleum reservoir simulation. Research in the petroleum engineering area was pioneer in its kind initiating a lot of sophisticated models [9, 10, 11, 36, 37, 38, 50] .

 

In the contaminant hydrology literature, a number of multiphase flow models have been presented. Faust [16] presented an isothermal two-dimensional finite difference simulator. It describes the simultaneous flow of water and NAPL under the saturated and unsaturated conditions. In this model, the gas is assumed as a stagnant phase with constant atmospheric pressure and thus the individual pressure equation for the gaseous phase is omitted. Osborne and Sykes [29] reported an isothermal twodimensional two-phase finite element model with the absence of the gas phase. Similarly, Parker et al. [33] and Kuppusamy et al. [27] developed a two-dimensional multiphase flow simulator involving three immiscible fluids: namely, air, water, and NAPL with the assumption of constant air phase pressure. Abriola and Pinder [1, 2] developed a two dimensional model that considers volatization and dissolution between phases. A similar model is presented by Corapcioglu and Baehr [13] . A subsequent extension was to incorporate hysteretic constitutive relations, which is done by Parker and Lenhard [32] and Lenhard and Parker [28] . Kaluarachchi and Parker [23, 24] applied a twodimensional finite element model named MOFAT-2D for three-phase, multi-component, isothermal flow and transport by allowing for interphase mass exchange but assuming gas phase pressure gradients are negligible. Sleep and Sykes [41] discussed a two-phase water-gas model with immobile NAPL phase, which also considers interphase mass transfer. As pointed out by Pinder and Abriola [34] , the extension of existing twodimensional models to third dimension, and solution of such three-dimensional problems presents a formidable task. But Faust et al. [17] developed a two-phase model based on a three-dimensional, finite difference formulation. Panday et al. [31] presented a threedimensional three-phase model, tested on highly nonlinear velocity fields. The STOMP (Subsurface Transport Over Multiple Phases) simulator developed by Pacific Northwest Laboratory [46, 47] simulates the transient coupled three-phase flow and heat transport using integral volume finite difference approach under the assumption that water phase is always present, and it is applicable in heterogeneous, multi-dimensional porous media. Pruess [36, 38] and Pruess et al. [37] developed the simulators TOUGH (Transport Of Unsaturated Groundwater and Heat) and its latest version TOUGH2, which are based on a finite difference scheme (MULKOM code) and simulate coupled heat and single component two-phase flow. Most of aforementioned models used the compositional modeling approach [1, 2, 13, 19, 20, 30, 42, 46, 47] . However, the applicability of aforementioned models to real situation is limited because of the intensive computational burden of solving the strongly coupled compositional equations (one equation for each component).

 

Even though many peoples have tried to improve efficiency and robustness of these models [14, 18, 19, 24, 42, 46, 47] , the models didn’t overcome some computational problems, which degrade the simulation performance for tough problems.

First, a phase switch jolts the system into possibly requiring more Newton iterations. This is because the primary variables are switched for the new state, but the starting values for the next iteration are not completely consistent with the new state of the system. Second, large storage requirements are imposed by the simulator due to the assembling of constraints in the global matrix. Finally, some switching criteria are very often used in these models, which might cause propagation of errors into the simulation domain when a phase configuration changes.

 

In contrast to the above compositional models, the fractional flow approach of governing equation can consider automatic change of phase configuration without some primary switching technique or switch criteria. However, the fractional flow approach has been employed by only a few peoples [4, 21] . NAPL simulator developed by Guarnaccia and Pinder [21] is based on a three-phase fractional approach model, accommodates hysteretic phenomena and allows NAPL to undergo dissolution and volatilization. This model applies the Hermite collocation finite element discretization, and the pressure and saturation equations are solved simultaneously by Picard iteration. Even though they used fractional flow approach, they didn’t consider commonly used boundary condition in hydrology. Also, boundary condition is treated like equivalent source and sink term, which might make convergence slow. Also, Binning and Celia [4] used fractional flow approach but developed only two-phase flow model (water-gas). They proved the better efficiency of fractional flow approach over the pressure based approach performing some numerical experiments. However they still encountered difficulty in implementing boundary condition. They considered only specific boundary condition rather than general boundary condition.

 

1.3. RESEARCH OUTLINE

 

In the current work, two and three-dimensional three phases numerical simulators (2DMPS and 3DMPS) are developed using the fractional flow based approach to simulate simultaneous movement of water, NAPL, and gas with general initial and boundary condition, respectively.  Even though some models have developed based on the fractional flow based approach [4, 21] , these models didn’t provide efficient and robust algorithm of implementing boundary condition and only considered specific boundary condition. However, in this study, the general boundary condition is considered, which consists of ten types. The efficient and robust algorithms of general boundary conditions are obtained and implemented in 2DMPS and 3DMPS. The first eight types are the combinations of two types of boundaries of individual phases, flux and Dirichlet conditions.  The other two types are the variable boundary conditions. Also the general initial conditions are made of eight combinations of two types of initial condition of individual phases, saturation and pressure. Therefore, no matter which type of initial and boundary conditions are specified among the eight types, it can be transformed and incorporated into initial and boundary conditions for total pressure, water saturation, and liquid saturation of the primary variables. Thus application of general boundary condition and initial conditions extend usefulness of 2DMPS and 3DMPS to the field while others [4, 21] have a limitation in applying boundary condition generally encountered in the field appropriately.

 

Another distinctive feature of this research is that 2DMPS and 3DMPS can handle phase configuration changes or phase appearance and disappearance in the solution domain without having to make assumptions and use troublesome variable switches [14, 18, 19, 24, 42, 46, 47] . Since the primary variables in fractional flow-based approaches exist throughout the solution domain regardless of whether the nonwetting phase is present or not, there is no need to specify a small, fictitious degree of nonwetting saturation where only the wetting phase is present, as is normally required with the traditional pressure-based simulators [23, 34] .  When the NAPL is absent in certain subregions, the governing equations for water saturation and liquid saturation automatically degenerate into identical equations, which yield the simulation of equality between water saturation and liquid saturation in the subregions. Therefore, 2DMPS and 3DMPS have the same advantages in dealing with automatic phase configuration change like other models [4, 21] on fractional flow approach, which are superior in efficiency and accuracy to other models [14, 18, 19, 24, 42, 46, 47] based on pressure based approach because pressure based approach used primary variable switch technique, switching criteria, introduction of some small fititious degree of NAPL where the NAPL is absent.

 

Since the saturation equations have the nature of hyperbolic equation for which some characteristic methods are well suited, we used some advanced Lagrangian-Eulerian method such as LEZOOMPC. Therefore, the pressure equation is solved by standard Galerkin finite element method, and two saturation equations can be simultaneously solved using LEZOOMPC or standalone upstream finite element method. Especially, the application of the Lagrangian-Eulerian method allows the 2DMPS and 3DMPS to have better efficiency and accuracy over the Eulerian approach. The theoretical backgrounds of the better performance of Lagrangian-Eulerian approach over the Eulerian approach have been well documented in other article [49] . Therefore, the models (2DMPS and 3DMPS) developed in this study have better efficiency and accuracy over the others [4, 21] using the Eulerian approach based on fractional flow approach.

 

In addition, multi-dimensional particle tracking algorithms central to LagrangianEulerian approaches were developed to generate more accurate results for linearly varying velocity field in both time and space.

 

 

1.4. FORMAT

 

This thesis is presented in the format of a collection of papers. It contains three papers (Suk et al., 2003a, 2003b, 2003c) submitted to three professional journals: Advances in Water Resources; International Journal for Numerical Methods in Engineering; and Journal of Hydrologic Engineering, ASCE. To maintain consistency, these papers are reorganized by renumbering the equations, figures, and tables of the original manuscripts. Most of the materials remain unchanged in each section of these papers. Chapter two was submitted to the Advances in Water Resources as “Multiphase flow modeling with general boundary conditions and phase configuration changes using fractional flow approach”. Chapter three presents “Particle tracking algorithm for the LagrangianEulerian finite element method under the transient conditions in multi-dimensions” which was submitted to the International Journal for Numerical Methods in Engineering. Chapter four was submitted to the Journal of Hydrologic Engineering as “Threedimensional multiphase flow simulation of three phases using the Lagrangian-Eulerian approach (LEZOOMPC)”. Chapter five summarizes the approach and results of this work and describes possible opportunities for future research.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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DEVELOPMENT OF 2- AND 3-D SIMULATOR FOR THREE-PHASE FLOW WITH GENERAL INITIAL AND BOUNDARY CONDITIONS ON THE FRACTIONAL FLOW APPROACH

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